In this section, we summarise a partial desingularisation of the reductive GIT quotient \(Y/\!/G\)  that is constructed by performing a canonical sequence of blow-ups along closed invariant subschemes determined by studying the dimensions of reductive stabiliser groups of semistable points [1]}. In [1]} this procedure was described when the stable locus was non-empty, in which case one has that semistability and stability coincide on the final blow-up. If one starts with a smooth projective scheme \(Y\) , the resulting blow-up \(\widetilde{Y}\)  is also smooth and projective and its GIT quotient \(\widetilde{Y}/\!/G\)  has only orbifold singularities and thus provides a partial desingularisation of \(Y/\!/G\) . One can also perform this blow-up procedure when then stable set is empty but the semistable set is non-empty; in this case, the procedure can terminate in several different ways, as we describe below.
